In this terminology, Theorem 1 asserts that every non-zero proper value of a CC-operator[completely continuous operator] has finite multiplicity. This result is not always helpful, for there exist CC-operators having no proper values at all:

– for each proper value α of L|Ec and proper value β of L|Eu, the proper value α/β — from the assumptions it follows that this latter collection of proper values consists of contracting proper values only.